In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} ...

states that any matrix is uniquely the sum of a diagonalizable matrix and a nilpotent matrix. Over an algebraically closed field, diagonalizable matrices...

See nilpotent matrix for more. In the factor ring Z / 9 Z {\displaystyle \mathbb {Z} /9\mathbb {Z} } , the equivalence class of 3 is nilpotent because...

triangular matrix are also 0, the matrix is called strictly (lower or upper) triangular. All finite strictly triangular matrices are nilpotent of index...

the zero matrix. Identity matrix, the multiplicative identity for matrices Matrix of ones, a matrix where all elements are one Nilpotent matrix Single-entry...

The matrix PX is idempotent. More generally, the trace of any idempotent matrix, i.e. one with A2 = A, equals its own rank. The trace of a nilpotent matrix...

i.e. when T is a square matrix (Nilpotent matrix) with complex entries, σ(T) = {0} if and only if T is similar to a matrix whose only nonzero entries...

{e^{4}+1}{2e}}\\\end{bmatrix}}.} A matrix N is nilpotent if Nq = 0 for some integer q. In this case, the matrix exponential eN can be computed directly...

The adjacency matrix can be used to determine whether or not the graph is connected. If a directed graph has a nilpotent adjacency matrix (i.e., if there...

inverse B − 1 {\displaystyle B^{-1}} and N {\displaystyle N} is a nilpotent matrix, then A D = [ B − 1 0 0 0 ] {\displaystyle A^{D}={\begin{bmatrix}B...

with M nilpotent, commuting with N and hence L. But then 0 = S2 − T2 = 2(L − M)(I + (L + M)/2). Since L and M commute, the matrix L + M is nilpotent and...

(the identity matrix), for any integer a between 0 and n inclusive. If N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form...

that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if and only if its...

1&0\\0&0\end{bmatrix}}.} For n ≥ 2, the matrix ring Mn(R) over a nonzero ring has zero divisors and nilpotent elements; the same holds for the ring of...

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension ⁠ 3 {\displaystyle 3} ⁠, playing a key role in...

Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as Nij = δi,j−1 (where δ is the Kronecker delta). The nilpotency...

zero matrix, corresponding to the dual number ε {\displaystyle \varepsilon } . Generally, if ε {\displaystyle \varepsilon } is a nilpotent matrix, then...

splitting of A. Gauss–Seidel method Jacobi method List of matrices Nilpotent matrix Successive over-relaxation Burden & Faires (1993, p. 404) Isaacson...

where D {\displaystyle D} is a diagonal matrix, N {\displaystyle N} is a nilpotent matrix, and [ D , N ] = 0 {\displaystyle [D,N]=0} , justifying...

similarly. A nilpotent element is an element a such that an = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix. A nilpotent element...

the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes λ 1 ≥ λ 2 ≥ … ≥ λ r , {\displaystyle...

Jordan–Chevalley decomposition which expresses a matrix as a sum of a diagonalizable and a nilpotent part. A Hermitian matrix has all real eigenvalues and can always...

then so is D and D−1(PAP−1) is equal to the identity plus a nilpotent matrix. But such a matrix is always invertible (if Nk = 0 the inverse of 1 − N is 1...

the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities...

triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains...

classification of Lie groups. Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and solvable Lie algebras. A Lie algebra...

complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix, and P {\displaystyle...

For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis...